Why isn't my FizBuzz code processing both if statements when they both match?

What you're trying to do is

if (a)
    ...
if (b)
    ...
else // if neigther a nor b
    ...

This is simply not possible. An else can only belong to a single if. You have to go with the slightly longer variant.

To avoid doing redundant evaluations of the modulo operator, you could formulate the loop body as

boolean fizz = i % 3 == 0;
boolean buzz = i % 5 == 0;

if (fizz) 
   System.out.print("Fizz");
if (buzz)
   System.out.print("Buzz");
if (!(fizz || buzz))
   System.out.print(i);

System.out.println();

---

Another one would be

String result = "";

if (i % 3 == 0)   result = "Fizz";
if (i % 5 == 0)   result += "Buzz";
if (result == "") result += i;

System.out.println(result);

If your only goal is to avoid using &&, you could use a double negation and DeMorgan's laws:

for(int i = 1; i <= 100; i++) {

    if(!(i % 3 != 0 || i % 5 != 0)) {
       System.out.println("FizzBuzz");
    } else if (i % 3 == 0) {
       System.out.println("Fizz");
    } else if (i % 5 == 0) {
       System.out.println("Buzz");
    } else {
       System.out.println(i);
    }
}

You can avoid && using the fact that i % 3 == 0 and i % 5 == 0 implies i % 15 == 0, as per RFC1337's answer.

Another solution is to use a switch on the remainder (mod 15, which is 5 times 3):

for(int i = 1; i <= 100; i++) {
    final int mod = i % 15;
    switch (mod) {
        case 0:
        case 3:
        case 6:
        case 9:
        case 12:
            System.out.print("Fizz");
            if (mod != 0) break;
        case 5:
        case 10:
            System.out.print("Buzz");
            break;
        default:
            System.out.print(i);
    }

    System.out.println();
}

Your first if statement is all alone.

So, your code hits the first statement, which is ONLY an if statement, and then goes on to the next, which is an if/else statement.

RosettaCode has a good example without using AND operators.

   int i;
   for (i = 0; i <= 100; i++) {
           if ((i % 15) == 0)
                   cout << "FizzBuzz" << endl;
           else if ((i % 3) == 0)
                   cout << "Fizz" << endl;
           else if ((i % 5) == 0)
                   cout << "Buzz" << endl;
           else
                   cout << i << endl;
   }